Congruence modular

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A Mal'cev condition (with 4-ary terms) for congruence modularity is given by [ Alan Day,
A characterization of modularity for congruence lattices of algebras.,
Canad. Math. Bull.
12
(1969)
167--173
MRreview ]

Another Mal'cev condition (with ternary terms) for congruence modularity is given by [ H.-Peter Gumm,
Congruence modularity is permutability composed with distributivity,
Arch. Math. (Basel)
36
(1981)
569--576
MRreview ]

Several further characterizations are given in [ Steven T. Tschantz,
More conditions equivalent to congruence modularity,
Universal algebra and lattice theory (Charleston, S.C., 1984)
Lecture Notes in Math.
1149
270--282
Springer
(1985)
MRreview ]

Changed: 9c58

Congruence permutability

Congruence n-permutable for n = 2 or n = 3.

Changed: 11c60

Congruence distributivity

Congruence distributive

An algebra is congruence modular (or CM for short) if its lattice of congruence relations is modular.

A class of algebras is congruence modular if each of its members is congruence modular.

Congruence modularity holds for many 'classical' varieties such as groups and rings.

Properties that imply congruence modularity

Congruence permutability

Congruence distributivity

Congruence modular

Properties that imply congruence modularity

Properties implied by congruence modularity